Simultaneous Characterization of Two Ultrashort Optical Pulses at Different Frequencies Using a WS2 Monolayer

The precise characterization of ultrashort laser pulses has been of interest to the scientific community for many years. Frequency-resolved optical gating (FROG) has been extensively used to retrieve the temporal and spectral field distributions of ultrashort laser pulses. In this work, we exploit the high, broad-band nonlinear optical response of a WS2 monolayer to simultaneously characterize two ultrashort laser pulses with different frequencies. The relaxed phase-matching conditions in a WS2 monolayer enable the simultaneous acquisition of the spectra resulting from both four-wave mixing (FWM) and sum-frequency generation (SFG) nonlinear processes while varying the time delay between the two ultrashort pulses. Next, we introduce an adjusted double-blind FROG algorithm, based on iterative fast Fourier transforms between two FROG traces, to extract the intensity distribution and phase of two ultrashort pulses from the combination of their FWM and SFG FROG traces. Using this algorithm, we find an agreement between the computed and observed FROG traces for both the FWM and SFG processes. Exploiting the broad-band nonlinear response of a WS2 monolayer, we additionally characterize one of the pulses using a second-harmonic generation (SHG) FROG trace to validate the pulse shapes extracted from the combination of the FWM and SFG FROG traces. The retrieved pulse shape from the SHG FROG agrees well with the pulse shape retrieved from our nondegenerate cross-correlation FROG measurement. In addition to the nonlinear parametric processes, we also observe a nonlinearly generated photoluminescence (PL) signal emitted from the WS2 monolayer. Because of its nonlinear origin, the PL signal can also be used to obtain complementary autocorrelation and cross-correlation traces.


Nondegenerate gradients
In the manuscript we use the common pulse retrieval algorithm (COPRA) algorithm developed by Geib et al. 1 to retrieve the fundamental pulse shapes from the FROG traces based on sum-frequency generation (SFG) and four-wave mixing (FWM). We extended the algorithm to also work with nondegenerate nonlinear signals and optimize for the SFG and FWM traces simultaneously. Here, we elaborate on the modification made with respect to the work of Geib et al. 1 For simplicity, we use the same formulation as the original paper.
Here,Ẽ n (ω) is the complex valued pulse envelope of the 775 nm and 1200 nm beams and E k (t) is its temporal counterpart. The time shifted 1200 nm pulse by the delay time τ is defined as, n,1200 ]. (S1) The spectral and temporal pulse shapes are related by following discrete Fourier transforms where ∆t and ∆ω are the time and frequency spacing respectively. Furthermore, the following relation between the DFT matrices is used: The COPRA retrieval algorithm calculates for each iteration the distance, Where S ′ is the experimental FROG trace and S(Ẽ) is the calculated trace from the current solutionẼ. S(Ẽ) for the nondegenerate nonlinear processes can be calculated in a similar S2 way as degenerate noncollinear FROG traces (even if the a collinear setup is used).
Now, to retrieve the next iteration complex electric field a single-gradient descent step is performed. In our case we want to optimize for the both the FWM and the SFG measurement trace.
As discussed in section 3 of the supplement information of 1 the derivation of ∇Z goes as, Here ∆S k = S ′ k − S k (Ẽ). Using the above formulas, we will derive the gradient steps for the SFG and FWM nonlinear processes. Note that we need to evaluate the gradient step for the two cases where we optimize the 1200 nm with the 775 nm as a reference and visa versa separately. So in total we derive four gradient steps: ∇Z SFG for optimization of the 1200 nm pulse,∇Z SFG for optimization of the 775 nm pulse, ∇Z FWM for optimization of the 1200 nm pulse and ∇Z FWM for optimization of the 775 nm pulse.

∇Z SFG for optimizing 1200 nm beam
Combining equations S3 and S6 to calculate the partial derivatives of equation S11.

∇Z FWM for optimizing 1200 nm beam
Combining equations S3 and S7 to calculate the partial derivatives of equation S11.